Modifications of Prony's Method for the Reconstruction of Structured Functions

The reconstruction and analysis of sparse signals is a common and widely studied problem in signal processing, for example in wireless telecommunication or power system theory. Hereby, most recovery methods exploit structures or special properties of the functions which are to be reconstructed. Particularly interesting are methods which aim to recover functions which possess a sparse representation in a given basis and use only a small set of sampling values. One of the most widely used methods is the so called Prony method, which is a deterministic method for the recovery of sparse exponential expansions. In recent year a generalization of Prony’s method for the reconstruction of spars expansion of eigenfunctions of certain linear operators has been introduced by Peter&Plonka in 2013. While some examples of suitable linear operators were given by Peter&Plonka, e.g., the shift operator as well as certain differential operators, the sample values needed for the reconstruction are not always accessible in practice. This leads to the following question. Can we find other suitable linear operators with meaningful structured functions as eigenfunctions and easily accessible sample values? Based on this question we investigate which structured functions can be recovered using only easily accessible sample values. Using the theory of one-parameter semigroups we derive a framework of so called generalized shift operators and their eigenfunctions, so-called generalized exponential sums, which covers all previously given examples for the generalized Prony method. Furthermore, we elaborate on the connection between generalized shift operators and linear differential operators and present a Prony based algorithm for the reconstruction of sparse generalized exponential expansion. Additionally, we present a new Prony based algorithm for the reconstruction of sparse expansions into orthogonal polynomials of length M using generating functions. Finally, we also consider the numerical analysis of the Prony method for generalized exponential sums and present a modified version of the ESPRIT algorithm and a sub-sampling based Prony method for the recovery of generalized exponential sums in the case of clustered frequencies.2021-12-2

The Magnus expansion and some of its applications

Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its applications to several physical problem

Study of special algorithms for solving Sturm-Liouville and Schrodinger equations

In dit proefschrift beschrijven we een specifieke klasse van numerieke methoden voor het oplossen van Sturm-Liouville en Schrodinger vergelijkingen. Ook de Matlab-implementatie van de methoden wordt besproken

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia

The 2nd International Conference on Mathematical Modelling in Applied Sciences, ICMMAS’19, Belgorod, Russia, August 20-24, 2019 : book of abstracts

The proposed Scientific Program of the conference is including plenary lectures, contributed oral talks, poster sessions and listeners. Five suggested special sessions / mini-symposium are also considered by the scientific committe

A spectral representation solution for electromagnetic scattering from complex structures

Significant effort has been directed towards improving computational efficiency in calculating radiated or scattered fields from a complex structure over a broad frequency band. The formulation and solution of boundary integral equation methods in commercial and scientific software has seen considerable attention; methods presented in the literature are often abstract, “curve-fits” or lacking a sound foundation in the underlying physics of the problem. Anomalous results are often characterized incorrectly, or require user expertise for analysis, a clear disadvantage in a computer-aided design tool. This dissertation documents an investigation into the motivating theory, limitations and integration into SuperNEC of a technique for the analytical, continuous, wideband description of the response of a complex conducting body to an electromagnetic excitation. The method, referred to by the author as Transfer Function Estimation (TFE) has its foundations in the Singularity Expansion Method (SEM). For scattering and radiation from a perfect electric conductor, the Electric-Field Integral Equation (EFIE) and Magnetic-Field Integral Equation (MFIE) formulations in their Stratton-Chu form are used. Solution by spectral representation methods including the Singular Value Decomposition (SVD), the Singular Value Expansion (SVE), the Singular Function Method (SFM), Singularity Expansion Method (SEM), the Eigenmode Expansion Method (EEM) and Model-Based Parameter Estimation (MBPE) are evaluated for applicability to the perfect electric conductor. The relationships between them and applicability to the scattering problem are reviewed. A common theoretical basis is derived. The EFIE and MFIE are known to have challenges due to ill-posedness and uniqueness considerations. Known preconditioners present possible solutions. The Modified EFIE (MEFIE) and Modified Combined Integral Equation (MCFIE) preconditioner is shown to be consistent with the fundamental derivations of the SEM. Prony’s method applied to the SEM poleresidue approximation enables a flexible implementation of a reduced-order method to be defined, for integration into SuperNEC. The computational expense inherent to the calculation of the impedance matrix in SuperNEC is substantially reduced by a physically-motivated approximation based on the TFE method. iv Using an adaptive approach and relative error measures, SuperNEC iteratively calculates the best continuous-function approximation to the response of a conducting body over a frequency band of interest. The responses of structures with different degrees of resonant behaviour were evaluated: these included an attack helicopter, a log-periodic dipole array and a simple dipole. Remarkable agreement was achieved